DTW (Dynamic Time Warping) 算法基于动态规划的思想，可以衡量两个长度不一致时间序列的相似度，由日本学者Itakura提出。

DTW

案例：

假设我们有三个时间序列，分别是

ts_a = [1,5,8,10,56,21,32,8]

ts_b = [1,5,8,10,23,56,21,32,8]

ts_c = [1,3,6,9,16,29,31,32,33]

ts_a与ts_b和ts_c的长度不一样，现在需要知道ts_a与ts_b和ts_c哪个更相似，通过观察，我们可以清楚的看出ts_a与ts_b的相似度更高。使用DTW相似度解决该问题的代码如下：

import sys
import numpy as np


def cal_dtw_distance(ts_a, ts_b):
    """Returns the DTW similarity distance between two 2-D
    timeseries numpy arrays.

    Arguments
    ---------
    ts_a, ts_b : array of shape [n_samples, n_timepoints]
        Two arrays containing n_samples of timeseries data
        whose DTW distance between each sample of A and B
        will be compared

    d : DistanceMetric object (default = abs(x-y))
        the distance measure used for A_i - B_j in the
        DTW dynamic programming function

    Returns
    -------
    DTW distance between A and B
    """
    d=lambda x, y: abs(x - y)
    max_warping_window = 10000

    # Create cost matrix via broadcasting with large int
    ts_a, ts_b = np.array(ts_a), np.array(ts_b)
    M, N = len(ts_a), len(ts_b)
    cost = sys.maxsize * np.ones((M, N))

    # Initialize the first row and column
    cost[0, 0] = d(ts_a[0], ts_b[0])
    for i in range(1, M):
        cost[i, 0] = cost[i - 1, 0] + d(ts_a[i], ts_b[0])

    for j in range(1, N):
        cost[0, j] = cost[0, j - 1] + d(ts_a[0], ts_b[j])

    # Populate rest of cost matrix within window
    for i in range(1, M):
        for j in range(max(1, i - max_warping_window),
                       min(N, i + max_warping_window)):
            choices = cost[i - 1, j - 1], cost[i, j - 1], cost[i - 1, j]
            cost[i, j] = min(choices) + d(ts_a[i], ts_b[j])

    # Return DTW distance given window
    return cost[-1, -1]

if __name__ == "__main__":
    # 案例：判断ts_a与ts_b和ts_c哪个更相似
    
    ts_a = [1,5,8,10,56,21,32,8]
    ts_b = [1,5,8,10,23,56,21,32,8]
    ts_c = [1,3,6,9,16,29,31,32,33]
    
    # 调用cal_dtw_distance计算dtw相似度
    dtw_ab = cal_dtw_distance(ts_a, ts_b)
    dtw_ac = cal_dtw_distance(ts_a, ts_c)
    
    print("ts_a与ts_b的dtw相似度为 %2.f，\nts_a与ts_c的dtw相似度为 %2.f。" % (dtw_ab, dtw_ac))
    
    if dtw_ab < dtw_ac:
        print("ts_a与ts_b 更相似！")
    else:
        print("ts_a与ts_c 更相似！")

ts_a与ts_b的dtw相似度为 13，
ts_a与ts_c的dtw相似度为 71。
ts_a与ts_b 更相似！

从执行结果可以看出，使用DTW相似度计算的结果与我们观察到的结果一致。